To determine which expression is equivalent to [tex]\(\left(4 x^5\right)^3\)[/tex], we need to apply the exponent rules correctly.
Let's break it down step by step:
1. Understand the expression:
[tex]\[\left(4 x^5\right)^3\][/tex]
This expression signifies that the entire term [tex]\(4 x^5\)[/tex] is raised to the power of 3.
2. Apply the power rule for exponents (which states [tex]\((a^m)^n = a^{m \cdot n}\)[/tex]):
- For the coefficient 4:
[tex]\((4)^3\)[/tex]
- For the variable term [tex]\(x^5\)[/tex]:
[tex]\((x^5)^3\)[/tex]
3. Calculate the coefficient:
[tex]\[4^3 = 4 \times 4 \times 4 = 64\][/tex]
4. Calculate the exponent of [tex]\(x\)[/tex]:
Using the rule [tex]\((a^b)^c = a^{b \cdot c}\)[/tex],
[tex]\[(x^5)^3 = x^{5 \cdot 3} = x^{15}\][/tex]
5. Combine the results:
[tex]\[\left(4 x^5\right)^3 = 4^3 \cdot (x^5)^3 = 64 x^{15}\][/tex]
Thus, the expression [tex]\(\left(4 x^5\right)^3\)[/tex] is equivalent to [tex]\(64 x^{15}\)[/tex].
Among the given options:
1. [tex]\(4 x^{5-3} = 4 x^2\)[/tex]
2. [tex]\(4^3 x^{5-3} = 64 x^2\)[/tex]
3. [tex]\(4 x^{5+3} = 4 x^8\)[/tex]
4. [tex]\(4^3 x^{5+3} = 64 x^8\)[/tex]
Clearly, none of the provided options exactly match [tex]\(64 x^{15}\)[/tex]. Therefore, the provided options appear to be incorrect for accurately matching [tex]\(\left(4 x^5\right)^3\)[/tex].
However, if the given answers were examined for common sub-expression forms without matching [tex]\(\left(4 x^5\right)^3\)[/tex] as inputs needing coordination, then:
- The correct choice would be derived with simplification exclusively. Re-evaluating the probable closest, correct simplification excluding above missed valid form, leverages correctly expressed:
[tex]\[4^3 x^{5 \cdot 3} = 64 \cdot x^{15}\][/tex]
Indeed, ensuring the validates coordination such:
- The completed solution aligns reasonably as [tex]\[\boxed{64 x^{15}} implying none exact mismatch resolving provided solutions.\][/tex]
